# Semiclassical Propagation: Hilbert Space vs. Wigner Representation

## TL;DR

This paper compares Hilbert space and Wigner representation approaches to semiclassical propagators, finding that the Hilbert space Herman-Kluk propagator remains the most effective despite the Wigner version's flexibility.

## Contribution

It provides a unified view of van Vleck and Herman-Kluk propagators across representations and evaluates the impact of Gaussian width flexibility in the Wigner approach.

## Key findings

- Numerical equivalence of propagators in both representations.
- Flexibility in Gaussian width does not improve accuracy.
- Hilbert space Herman-Kluk remains the preferred method.

## Abstract

A unified viewpoint on the van Vleck and Herman-Kluk propagators in Hilbert space and their recently developed counterparts in Wigner representation is presented. It is shown that the numerical protocol for the Herman-Kluk propagator, which contains the van Vleck one as a particular case, coincides in both representations. The flexibility of the Wigner version in choosing the Gaussians' width for the underlying coherent states, being not bound to minimal uncertainty, is investigated numerically on prototypical potentials. Exploiting this flexibility provides neither qualitative nor quantitative improvements. Thus, the well-established Herman-Kluk propagator in Hilbert space remains the best choice to date given the large number of semiclassical developments and applications based on it.